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Properties

Label 1-1287-1287.112-r0-0-0
Degree $1$
Conductor $1287$
Sign $-0.372 + 0.927i$
Analytic cond. $5.97680$
Root an. cond. $5.97680$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.406 − 0.913i)2-s + (−0.669 + 0.743i)4-s + (−0.406 + 0.913i)5-s + (0.207 − 0.978i)7-s + (0.951 + 0.309i)8-s + 10-s + (−0.978 + 0.207i)14-s + (−0.104 − 0.994i)16-s + (−0.809 + 0.587i)17-s + (−0.951 − 0.309i)19-s + (−0.406 − 0.913i)20-s + (0.5 + 0.866i)23-s + (−0.669 − 0.743i)25-s + (0.587 + 0.809i)28-s + (0.978 + 0.207i)29-s + ⋯
L(s)  = 1  + (−0.406 − 0.913i)2-s + (−0.669 + 0.743i)4-s + (−0.406 + 0.913i)5-s + (0.207 − 0.978i)7-s + (0.951 + 0.309i)8-s + 10-s + (−0.978 + 0.207i)14-s + (−0.104 − 0.994i)16-s + (−0.809 + 0.587i)17-s + (−0.951 − 0.309i)19-s + (−0.406 − 0.913i)20-s + (0.5 + 0.866i)23-s + (−0.669 − 0.743i)25-s + (0.587 + 0.809i)28-s + (0.978 + 0.207i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.372 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.372 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.372 + 0.927i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ -0.372 + 0.927i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1302007329 + 0.1926505227i\)
\(L(\frac12)\) \(\approx\) \(0.1302007329 + 0.1926505227i\)
\(L(1)\) \(\approx\) \(0.6088773982 - 0.1459026848i\)
\(L(1)\) \(\approx\) \(0.6088773982 - 0.1459026848i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.406 - 0.913i)T \)
5 \( 1 + (-0.406 + 0.913i)T \)
7 \( 1 + (0.207 - 0.978i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (-0.951 - 0.309i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.978 + 0.207i)T \)
31 \( 1 + (-0.994 - 0.104i)T \)
37 \( 1 + (0.951 - 0.309i)T \)
41 \( 1 + (0.207 + 0.978i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.743 - 0.669i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.743 - 0.669i)T \)
61 \( 1 + (0.104 + 0.994i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (-0.587 - 0.809i)T \)
73 \( 1 + (-0.951 + 0.309i)T \)
79 \( 1 + (-0.913 + 0.406i)T \)
83 \( 1 + (-0.994 + 0.104i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.406 + 0.913i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.62491371017779762037845749209, −19.934745967214168921962099048261, −18.97767814507239885348008277678, −18.53216541775856412936597626928, −17.53809335986471872636083912353, −16.96317947571492516797607524309, −16.06380286997547596930675654535, −15.59393704525982583061964794065, −14.86046156525073106317149227246, −14.039103279543483850514454577090, −12.99429620186490578788425926923, −12.44242739573341115879702024255, −11.419421993701367450292307243885, −10.51448403159346536704042525723, −9.37225672748608387922758922905, −8.79887631420781327658038088492, −8.32026182284191412707867875209, −7.36650949041694314928191836245, −6.39706420422868384422889205570, −5.58788814427815703445068196772, −4.76716678311668473944132910208, −4.15519326916169654324164022928, −2.54493362196582323781355766836, −1.42713663497175706589668339067, −0.11271839090401881355344584360, 1.29849796187478497478558795528, 2.343303403293091370300758779942, 3.28222595191157763220041607041, 4.05929033700175462852902898298, 4.7438038163300403322759099865, 6.3061589524016002956679119796, 7.16404311780691790851053929327, 7.866100965359802066375546790265, 8.70062708301921558733197970996, 9.73327845388915634518792200690, 10.51390046327331253784457089805, 11.07399015475405652825334904755, 11.5670682879449682444205928799, 12.8001461866031840637011761297, 13.32412947968145365572287631429, 14.269071241848979392911216496339, 14.94315361561784088470972889494, 16.0039790583396955129327911882, 16.96335185060522483903957452806, 17.602404347789536511272147179355, 18.24249806172912664312746122942, 19.12735487922109019150887806410, 19.80186604472213990817720391088, 20.13457976349769202808518192690, 21.47268605459536682833698561541

Graph of the $Z$-function along the critical line